Ziegler closures of some unstable tubes

TL;DR

This paper studies the structure of modules in the Ziegler closure of certain tubes in module categories over finite-dimensional algebras, extending previous results and providing new conditions for indecomposability.

Contribution

It improves on Krause's results by showing inverse limits along corays are indecomposable and describes modules over iterated one-point extensions of valuation domains.

Findings

Inverse limits along corays in stable tubes are indecomposable.

Descriptions of finitely presented modules over iterated one-point extensions.

A sufficient condition for the $k$-dual of a $ ext{Sigma}$-pure-injective module to be indecomposable.

Abstract

We describe the modules in the Ziegler closure of ray and coray tubes in module categories over finite-dimensional algebras. We improve slightly on Krause's result for stable tubes by showing that the inverse limit along a coray in a ray or coray tube is indecomposable, so in particular, the inverse limit along a coray in a stable tube is indecomposable. In order to do all this, we first describe the finitely presented modules over and the Ziegler spectra of iterated one-point extensions of valuation domains. Finally we give a sufficient condition for the $k$-dual of a $\Sigma$-pure-injective module over a $k$-algebra to be indecomposable.